In classical logic if either A or B is false then "A and B" is false. But in natural language it's often the case to hear someone say "that's only partially true" or "that's partly false" as a retort to hearing a sequence of statements some of which are false (often followed by an explanation what part was false).

One way to go at this is if there are truth-value gaps in the logic, i.e. a many-valued logic. However, the existence of gaps alone doesn't entail that the conjunction of true and false is necessarily going to be a gap. In fact, many many-valued logics have it that "true and false = false", e.g. that's the case in (strong) Kleene, or LP, or Łukasiewicz's logic. Even in Belnap's four-valued logic, although it has notions of "neither true nor false" and "both true and false", the conjunction of something that's (just) true with something that's (just) false... is again (just) false. In a comment below, someone suggested fuzzy logics, which are continuously truth-valued (over [0, 1]), but from what I can tell the conjunction of 0 and 1 is 0 in this family: the "conjunctive truth function * [...] must be commutative [and have] (0*x)=0".

I see from a 2003 paper of Humberstone, which I've only looked at very briefly that one approach is to have some kind of operator other than the actual truth valuation defined. (This seems to be a kind of "alternative valuation" of sorts. The paper, which is quite long, doesn't seem incredibly cited, so either the problem I'm talking about about is obscure, or Humberstone's proposed solution to is so...)

So have any logics been proposed in which the conjunction of true and false is a gap, rather than false? Or if not, is there any other way to formalize this notion that can capture that a conjunction of true and false is "partly true", besides Humberstone's operator approach, which seems a bit unsatisfactory as it uses a different "scale of truth" for this sort of evaluation than the main truth-valuation used in the logic?

(N.B: I suspect this comes down to some "deeper" issues like what can be reasonably called a conjunction in a logic and what is "false", because in a [semi]lattice with false as the bottom element, if conjunction is taken as the greatest lower bound on the [semi]lattice, it's inescapable that "false and anything = false".)

  • "that's only partially true" or "that's partly false" are modelled in Zadeh's Fuzzy Logic (which is based on Lukasiewicz essentially), just not with a conjunction of truth and falsity, but with the truth value introduced as having a value within the range of [0, 1] (and every value in between). – k-wasilewski Apr 4 at 16:00
  • Any reason for the DV? @k-wasilewski: and what does "Zadeh's" logic say about the conjunction of 1 and 0? – Fizz Apr 4 at 17:18
  • There's the en.wikipedia.org/wiki/Catu%E1%B9%A3ko%E1%B9%ADi I think ypu have misunderstood the impact of four valued logic, which can affect computer evaluation & electronics much more profoundly than you imply (eg allowing strange loops, & tangled hierarchies).Quantum logic is important, and should be related to signal loss through Shannon Entropy, and reality as information flow (ie Carlo Rovelli's picture). I can't see why the DV either, seems a perfectly relevent and well-asked question. – CriglCragl Apr 4 at 19:04
  • IMHO "that's only partly true" actually has nothing to do with formal logic at all. It usually indicates an informal fallacy such as equivocation (which, if you really want, you can turn back into a formal fallacy such as the fallacy of four terms, but that's the "false" part, not the "partly" part). – Kevin Apr 4 at 20:23
  • @Kevin: well, obviously getting only the "false" part is very easy in practically any logic I could remember... – Fizz Apr 4 at 21:33

One way to go besides Humberstone's approach might start from truth-maker semantics as developed by Kit Fine and other. Let's fix some language of standard propsitional logic and let's take models to be complete join semi-lattices induced by a partial order ≤. Then we can define a relation of exact verification and a relation of exact falsification analagous to some semantics for FDE, with the exception of the verification clause for conjunction: A point s verifies a conjunction A & B iff s is the supremum of points r, t, where r verifies A and t verifies B.

Based on this definition we can define a relation of conjunctive parthood between propositions (sets of points): Proposition p is a conjunctive part of proposition q iff every verifier of p has a ≤-successor among the verifiers of q and every verifier of q has a ≤-predecessor among the verifiers of p.

Now we can say that proposition p is partially true at a point s if it has a conjunctive part that is verified by s; and that p is partially false at s, if it has a conjunctive part that is falsified by s.

  • I see Fine (2017) cites Yablo's Aboutness as part of this field. I've already discovered that Yablo in that book cites approvingly Humberstone (2003). (He also mentions Russel's criticism of the whole endeavor though, which is what might have earned me the DV, maybe.) – Fizz Apr 4 at 23:42
  • To put this in a slogan, Fine (semantically) distinguishes between containment and entailment. What Humberstone does is to evaluate claims of containment in the same logic (via an operator). – Fizz Apr 5 at 0:43
  • That's interesting. I see Fine's semantics as an attempt to spell out a somewhat neglected tradition of providing a semantics for FDE, which dates back to work of van Fraassen in the 1960s. Sounds like Humberstone might have followed the same line. The more since Humberstone in his 2003 piece cites the relevant paper by van Fraassen. – sequitur Apr 5 at 21:19

This is only partial answer, but interestingly enough a sort-of-conjunction that would make "true and false" be "neutral" has been considered on 3-valued tables (and in fact [finite] multi-valued/fuzzy logics in general). It's however (obviously) not compatible with the Boolean one. From "A map of dependencies among three-valued logics" by D. Ciucci & D. Dubois.

2.1.3. t-operators

This is another generalization of t-norms on pre-ordered sets [ref42].

Definition 3 [ref43]. A binary operator * on a finite scale {F < x1 < . . . < T} is named t-operator if it is associative, commutative, such that F * F = F, T * T = T and it satisfies 1-smoothness: if xi * xj = xk then { xi-1 * xj, xi * xj-1 } ⊆ { xk-1, xk }.

(I've fixed the def looking up the original paper; it's garbled in this review by some missing symbols... but otherwise they give a somewhat more obvious form than in the original, albeit equivalent. Basically the t-operator is non-decreasing in each place and only goes "up by one" at the most, on the chain, in each argument.)

Proposition 4. On three-valued scales, there are only five t-operators: Kleene and Łukasiewicz conjunctions and disjunctions and the aggregation operator in Table 3.

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This has quite a trivial structure on the 3-valued chain: it gives the gap value everywhere except for F * F and T * T.

The peculiarity on the operator in Table 3 is that it does not generalize Boolean connectives: F and T yield the third value N. In fact it is easy to see that it is the operation med(x, y, N) computing the median between x, y and N. It is known to be the only associative operation between ∧ and ∨, and a special case of Sugeno Integral.

  • [ref42] M. Mas, G. Mayor, J. Torrens, t-Operators, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 7 (1) (1999) 31–50.

  • [ref43] M. Mas, G. Mayor, J. Torrens, t-Operators and uninorms on a finite totally ordered set, International Journal of Intelligent Systems 14 (1999) 909–922.

So if one wants a 3-value "conjunction" to still be commutative, associative, and have this kind of "part-truth" extraction, this is the only option... It's of course obtainable as a derived operator on any functionally complete 3-valued logic.

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